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| '''Semi-grand ensembles''' are used in [[Monte Carlo]] simulation of [[mixtures]]. In these ensembles the total number of molecules is fixed, but the composition can change.
| | == General Features == |
| == Canonical ensemble: fixed volume, temperature and number(s) of molecules == | | Semi-grand ensembles are used in Monte Carlo simulation of mixtures. |
| We shall consider a system consisting of ''c'' components;. | | |
| In the [[Canonical ensemble|canonical ensemble]], the differential | | In this ensembles the total number of molecules is fixed, but the composition can change. |
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| | == Canonical Ensemble: fixed volume, temperature and number(s) of molecules == |
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| | We will consider a binary system;. |
| | In the Canonical Ensemble, the differential |
| equation energy for the [[Helmholtz energy function]] can be written as: | | equation energy for the [[Helmholtz energy function]] can be written as: |
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| : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>, | | : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>, |
| where: | | where: |
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| *<math> A </math> is the [[Helmholtz energy function]] | | *<math> A </math> is the [[Helmholtz energy function]] |
| *<math> \beta := 1/k_B T </math> | | *<math> \beta \equiv 1/k_B T </math> |
| *<math> k_B</math> is the [[Boltzmann constant]] | | *<math> k_B</math> is the [[Boltzmann constant]] |
| *<math> T </math> is the absolute [[temperature]] | | *<math> T </math> is the absolute temperature |
| *<math> E </math> is the [[internal energy]] | | *<math> E </math> is the internal energy |
| *<math> p </math> is the [[pressure]] | | *<math> p </math> is the pressure |
| *<math> \mu_i </math> is the [[Chemical potential|chemical potential]] of the species <math>i</math> | | *<math> \mu_i </math> is the chemical potential of the species "i" |
| *<math> N_i </math> is the number of molecules of the species <math>i</math> | | *<math> N_i </math> is the number of molecules of the species "i" |
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| == Semi-grand ensemble at fixed volume and temperature == | | == Semi-grand ensemble at fixed volume and temperature == |
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| Consider now that we wish to consider a system with fixed total number of particles, <math> N </math> | | Consider now that we want to consider a system with fixed total number of particles, <math> N </math> |
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| : <math> \left. N = \sum_{i=1}^c N_i \right. </math>; | | : <math> \left. N = N_1 + N_2 \right. </math>; |
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| but the composition can change, from thermodynamic considerations one can apply a [[Legendre transform]] [HAVE TO CHECK ACCURACY] | | but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY] |
| to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>. | | to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>. |
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| * Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>
| | # Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> |
| | | : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math |
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| : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_i d N_i; </math>
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| : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_i-\mu_1) d N_i; </math>
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| or,
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| : <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math> | |
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| where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>.
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| * Now considering the thermodynamic potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>
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| :<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N -
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| \sum_{i=2}^c N_i d \left( \beta \mu_{i1} \right).
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| </math>
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| == Fixed pressure and temperature ==
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| In the [[isothermal-isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write:
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| :<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>
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| where:
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| * <math> G </math> is the [[Gibbs energy function]]
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| == Fixed pressure and temperature: Semi-grand ensemble ==
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| Following the procedure described above one can write:
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| :<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,
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| where the ''new'' thermodynamic potential <math> \beta \Phi </math> is given by:
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| :<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
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| - \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
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| </math> | |
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| == Fixed pressure and temperature: Semi-grand ensemble: partition function ==
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| In the fixed composition ensemble one has:
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| :<math> Q_{N_i,p,T} = \frac{ \beta p }{\prod_{i=1}^c \left( \Lambda_i^{3N_i} N_i! \right) } \int_{0}^{\infty} dV e^{-\beta p V } V^N
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| \int \left( \prod_{i=1}^c d (R_i^*)^{3N_i} \right) \exp \left[ - \beta U \left( V, (R_1^*)^{3N_1} , \cdots \right) \right].
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| </math>
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| ==References==
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| <references/>
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| ;Related reading
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| *[http://dx.doi.org/10.1063/1.3677193 Yiping Tang "A new method of semigrand canonical ensemble to calculate first-order phase transitions for binary mixtures", Journal of Chemical Physics '''136''' 034505 (2012)]
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| [[category: Statistical mechanics]]
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